Trefftz method in solving the inverse problems
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- Dylan Stone
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1 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola ff mho solvg h vs poblms Ksof Gsa Klc Uvs of cholog Al.. -lca P.P Klc Pola -mal: sof@gsa.pl /7
2 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola ff mho has b ow sc h wo of ff 96 *. A appoma soluo of a poblm s a la combao of fucos ha sasf h govg ffal la quao o such o ha s possbl o b cov o such a fom. h uow coffcs a m fom h coos of appoma fulfllg h boua a al coos fall havg a fom of a ssm of algbac quaos. * E. ff E Ggsu um R sch Vfah. Pocgs of h Iaoal Cogss of Appl Mchacs 3 37 Zuch 96. /7 Ksof Gsa - ff mho solvg h vs poblms
3 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Gall ff bass fall o wo boa classs F-ff bass bas o fuamal soluos a -ff bass whch a usuall bu o alwas oba b spaao of vaabls pola a Casa cooa ssms. I Całows ps wo oh mhos o of whch bas o vlopg fuco alo ss s paculal smpl a ffcv. W wll focus ou ao o -ff bass. I ghs a s -compl fucos hav b us o f appoma soluos of BVP also wh h us of FEM. Howv o IBVP w vsga. Rsachs us o g of m vaabl a as a sul h cos h Hlmhol p quao. hfo h fs -compl fucos w fou fo Laplac quao bhamoc o a fo uc wav quao Hlmhol p q. MJ M.J. Całows A. Fącowa Ha fucos a h applcao o solvg ha couco a mchacal poblms. W. Polch Poańs Poań. Polsh Ksof Gsa - ff mho solvg h vs poblms 3/7
4 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Dug h las as ff mho has b appl o f appoma soluos of h vs poblms. Up o ow h followg vs poblms hav b cos: - boua valu mao vs poblms - maal pops mao vs poblms - soucs mao vs poblms. A fs a global appoach.. loog fo a appoma soluo h whol oma was appl wh goo suls fo smpl gom a al-boua coos. Howv a ga mao of mo compl poblms of mahmacal molg cao b solv whou vso of h aa Ω o subgos lms. FEM wh ff fucos as al fucos s h us. Ksof Gsa - ff mho solvg h vs poblms 4/7
5 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola ff fucos fo som la ffal quaos whou m * Laplac quao: u D Ω { R Im } { q φ ;... q } q 3D Ω P cos θ D Ω 3D Ω P q cosθ Ruc wav quao: - assoca Lg fucos u u { J J cos θ J s θθ... } q qφ { P cosθ ;... q } π J / * A.P. Zlńs O al fucos appl h gal ff mho. Av. Eg. Sofwa Ksof Gsa - ff mho solvg h vs poblms 5/7
6 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola ff fucos fo som la ffal quaos wh m Ha couco quao: u u Ha couco quao: D Ω u [ ]...!! v D Ω!! m v v V m Bam vbao quao * : 4 4 u u S S S S! ! S S a fo 3 6/7 Ksof Gsa - ff mho solvg h vs poblms M.J. Al-Khab K. Gsa A. Macąg h mho of solvg polomals h bam vbao poblms. J. ho. Appl. Mch
7 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola ff Mhos ff Mhos hav o cv a pcs fo alhough hs molog has ha w accpac. Ha s fo of wha s ma b a ff Mho s: Gv a go of a Eucla spac o som paos of ha go a ff Mho s a pocu fo solvg al boua valu poblms of paal ffal quaos o ssms of such quaos o such go usg soluos of ha ffal quao o s ao f s subgos. Wh ff Mho s cocpual hs ma clus ma of h basc poblms cos umcal mhos fo paal ffal quaos a bcoms a fuamal cocp of ha subc. I. Ha ff mho: A gal ho. Num. Mh. Paal Dff. Eq Ksof Gsa - ff mho solvg h vs poblms 7/7
8 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Ic ff Mhos Cos a la ff q Lu Ω wh valus of uu pscb o a u/q q pscb o. a a pas of Ω o a clu Ω. u q N * * P u P a u a u P * P q P P a P u q { * u } * Rsuals: R a u P u P fo P * R a q P q P fo P --complfucos. Collocao mho: suals a h pos P plac o a u P a foc o vash Kaf a. Las-squa-mho: F a R m Kaf α R a u P Ksof Gsa - ff mho solvg h vs poblms α - wghg paam 8/7
9 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Ic ff Mhos Cos a la ff q Lu Ω wh valus of uu pscb o a u/q q pscb o. a a pas of Ω o a clu Ω. 3. Gal mho fomulao: F a qr ur q a u - wghg paam u P Kaf 4. Mof ff fomulao -ff appoach I hs cas h -compl fucos a bul usg fuamal soluo fo h q Lu. h f.s. s a fuco of PQ wh P bg a gal po of Ω a Q sag fo al souc plac o h maga boua suoug h al boua. h u N u * * P u P a u P a * Q P q P q P P a q P a Ksof Gsa - ff mho solvg h vs poblms u 9/7
10 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Dc ff Mhos h c ff fomulao s bas o h boua gal quao. Dals fo h Laplac q: h acl E. Ka N. Kama ff mho: a ovvw. Av. Eg. Sofwa Ksof Gsa - ff mho solvg h vs poblms /7
11 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Global appoach BVP Cos a saoa ha couco poblm a hollow cl: p R H R b a M N b b m m m w h c θ b π π H C a π s Eac soluo: b b H H b H a F H b H F H C π π π π π ; ; s ; I K K I F Appoma soluo has b oba wh 8 s h a w - fucos [ ] / fucos [ ] /!! h [ ] / l a h w /7 Ksof Gsa - ff mho solvg h vs poblms Eac Appo a. m b. m H. m!! l h w
12 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola..6 A squa 4 lms vs BVP O h ss of h squa h Dchl cos boua coos a pscb accoac wh h accua soluo * : Usg oms L a H f H L f L δ H L L h appoma L a ac soluos a compa. f H. f H.6..8 c f H π π sh π sh δ H H f L 6 δ L 49 f H 84 δ H 85 H..8. a b f L.4. δ f L. δ Dsac fom h boua wh uow coo δ vsus oms f H a f L fo a 3 b a c pos wh masu mpau f L δ.. δ Appoma soluo fo pos wh masu mpau * M.J. Całows A. Fącowa Ha fucos a h applcao o solvg ha couco a mchacal poblms. W. Polch Poańs Poań. Polsh /7 Ksof Gsa - ff mho solvg h vs poblms
13 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola FEM fo h IHCP: o FEM wh coo of cou of mpau h commo os of lms. o No mpau cou a a po bw lms. 3 o Nolss FEM. Isa ach f lm h mpau s appoma wh h la combao of h ff fucos. h uow coffcs of h combao a calcula fom h coo of mmg h fucoal ha scbs h ma-squa fg of h appoma mpau fl a lm o h boua a al coos. Moov h gc gulasao s us o mpov h appoma soluo.. o mms fc of g sspao bw lms oumcal op pouco bg a sul of scou of ha flu bw lms. Ksof Gsa - ff mho solvg h vs poblms 3/7
14 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola A squa 4 lms mpau scouous bw lms q p q q Q α Obcv fucoal: Accuac of h appoma soluo q q Q [ ] % / Ω Ω δ Ω Ω FEM couous FEM scouous N V a α. α α α. α α δ Nom as a fuco of sac fo δ δ δ al fucos. Ipu aa a accua. 4/7 Ksof Gsa - ff mho solvg h vs poblms
15 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola h gc gulasao fo IHCP * I o o f a appoma soluo o mms a ha flu ump o fc of g sspao o fc of umcal op pouco bw lms. h ms h fucoal a mm o f coffcs h fomula scbg. h a: h ha flu ump q& q& q & h fc of g sspao q& q& h fc of umcal op pouco q& l q& l * M.J.Całows A. Fącowa K. Gsa Soluo of asaoa vs ha couco poblm b mas of ff o-couous mho. I. J. Ha Mass asf /7 Ksof Gsa - ff mho solvg h vs poblms
16 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola A squa h FEM appoachs h gc gulasao A IHCP a squa s cos *. Coos: h h 3 cos s b... 8 h Accua al mpaus Accua soluo: cos s b δ L Dsac vsus lav o fo mmsao of a ha flu b op pouco c g sspao fo couous scouous olss FEM A. Macąg ff fucos fo som c a vs poblms of mchacs. Polcha Śwęosa Klc 9 Polsh. 6/7 Ksof Gsa - ff mho solvg h vs poblms
17 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola m-spac f lms wh ϕ v h -h al ff fuco N av... N hc h mpau h -h lm: N ϕ bg a la combao of h ff fuco. m-spac lms couous os δl D D Rlav o D / D m-spac lms scouous os Ksof Gsa - ff mho solvg h vs poblms 7/7
18 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola A squa o-saoa IHCP FEM * Ha asf q. Δ / Coos: h Ial mpaus δb { } { } h h3 h ac soluo *K. Gsa R. Lśwsa Dff f lm appoachs fo vs ha couco poblms. Iv. Pobl. Scc a Eg. 8: 3 7 8/7 Ksof Gsa - ff mho solvg h vs poblms
19 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Obcv fucoal D h D f J IR I h h 3 b δ ha flu ump J S l l l l op po. ump J E g ss. ump 9/7 l l l l g p J RE
20 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Iaccua pu aa h m-spaal oma s v o 4 lms. h masums a smula fom h ac soluo a sub wh a os wh omal sbuo o ga ha 5% of h ac valu. I o o oba goo suls h pu aa hav b smooh wh h us of 8 ff fucos. δlδ L h lav o δl vsus h sac δ b fo accua a accua smooh pu aa fo olss FEM whou gc gulasao /7 Ksof Gsa - ff mho solvg h vs poblms
21 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola h appoma mpau accuac h olss FEM fo ff fucos Ksof Gsa - ff mho solvg h vs poblms /7
22 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola h appoma mpau accuac h olss FEM fo 5 ff fucos Ksof Gsa - ff mho solvg h vs poblms /7
23 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Rmas h s poblms wh gc gulaao la o v goo suls fo h all h mhos. h bs o appl sm o b h olss FEM. h lav o os o c % v om δh fo accua a smooh pu aa a al fucos. h ga umb of -fucos h b suls o obas. Smoohg h accua aa wh h us of ff fucos las o suls compaabl wh hos oba wh accua pu aa. I h cas of a c poblm all h mhos la o goo suls. Ksof Gsa - ff mho solvg h vs poblms 3/7
24 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Op poblms:. A p of m-spac lm ps o h p of govg quao bcaus h lgh of h m s of h lm shoul pobabl p o h sgal popagao vloc.. Fo h HC poblms a vloc of mpau popagao p shoul b la o h mpau masum accuac. 3. Gall FEM bg f lms ca b us. Howv h s ps o h umb of al fucos. 4. I h placs of accumulao of h vsga phomo h lms shoul b coca spac a m. Fa fom such placs h m-spac lms ca b ga m a spac. 5. A appoma soluo of a c vs poblm sms o b of b qual f h coos a pu aa a fomula h sam subspac of h spac ga b h -compl fucos. Ksof Gsa - ff mho solvg h vs poblms 4/7
25 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Op poblms: 6. h olss FEM sms o b a sg mho o am ms of soluo scous a h gs bw h lms vsus h s of h lms a msos of h subspac ga b -fucos. 7. I ou su o -fucos applcao small umb of pos wh pu aa al sposs l o goo suls. Also h compl aa g lac of al coo o h boua coos ow ol o a pa of h boua la o goo compaabl wh accua suls. 8. Rgulasao wh h us of omal vav ump o h bos bw m-spac lms sms o b sg o vsga. 9. I h cas of ohomogous ff quao a a of appomag h gh s of h q. wh h -fucos sms o smplf h vsgao. Ksof Gsa - ff mho solvg h vs poblms 5/6
26 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola Mahmacas a gs I h acl of Z.C. L.. Lu H.. Huag A.H.-D. Chg ff Collocao a Oh Boua Mhos A Compaso. Num. Mh. Pa.Dff. Eq I hav fou h followg ma: I s of s o po ou ha som of h Ms a vlop h gg commu [E. Ka N. Kama ff mho: a ovvw. Av. Eg. Sofwa ] whl ohs h mahmacal commu [Z.C. L Comb mhos fo llpc quaos wh sgulas facs a fs. Kluw Acamc Publsh Doch 998 ]. As comm a acl o h hso of boua mhos [A.H.-D. Chg D.. Chg Hag a al vlopmof boua lm. Eg. Aal. Bou. Elm ] h vlopms h wo commus h appl mahmacs a h gg sms o a paalll o ach oh almos vo of a coss caos alhough ma chqus hav much commo a coss-flao s. I woul b usful o uf h sus of h wo commus a o smula coss-flao a cao. Ksof Gsa - ff mho solvg h vs poblms 6/7
27 IP-A Ivs Poblms: vlopms ho a applcaos Fbua 9- Wasaw Pola ha ou fo ou ao Ksof Gsa Klc Uvs of cholog Al.. -lca P.P Klc Pola -mal: sof@gsa.pl Ksof Gsa - ff mho solvg h vs poblms 7/7
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